Abstract
In this paper, we examine two opposite scenarios of energy behaviour for solutions of the Euler equation. We show that if u is a regular solution on a time interval [0, T) and if u ∈ LrL∞ for some , where N is the dimension of the fluid, then the energy at the time T cannot concentrate on a set of Hausdorff dimension smaller than . The same holds for solutions of the three-dimensional Navier-Stokes equation in the range 5/3 < r < 7/4. Oppositely, if the energy vanishes on a subregion of a fluid domain, it must vanish faster than (T − t)1−δ, for any δ > 0. The results are applied to find new exclusions of locally self-similar blow-up in cases not covered previously in the literature.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
